I don't think the few gov workers who were furloughed made much difference in the search effort. they probably are not going to have all that much to do with the actual searching anyway.
Well, that depends. With how many searches in NPS areas have you been involved? Offhand, I can recall five operations with which I have had personal experience (one for more than two months total) and I can tell you that in all these cases the presence and abilities of the staff helped considerably.
One thing about search operations, especially those that persist for more than a couple of days, is that they can become quite complex. Good detailed knowledge of the terrain and features of the search area is obviously critical. I know nothing of the knowledge, skills, and abilities of the individuals who work there, but I would be very surprised if there isn't at least one or two people who know the area very well, and are able to provide real help, either out ground pounding or advising in base camp (and that is not at all trivial).
And let's consider the terrain- it is to a good extent what is known as a'a lava - jagged bits of volcanic rock pushed and piled into chaotic jumbles. You don't hike through this country. It is more like continuous, low level rock climbing (AKA bouldering). You often have lava caves and tubes of varying extent which surely have a high priority in the search. I note that they have a couple of cavers participating in the effort and that is a very good idea. In this type of terrain, intimate knowledge is even more important than in more "normal" country.
At some point in this operation, it is likely that they would need to call in some fancy technology - heat sensors and the like which would detect the presence. Requests from a federal agency, sometimes from high up, can facilitate the appearance of "non-normal" items, like drones, for example. I would think they would be ideal for a situation like this.
And finally, there is good, old fashioned, mundane logistics. You want to have people out in the field searching. Once the search goes beyond day, you have got to get busy feeding, sleeping, supporting, resupplying, and relieving the folks in the field. Those on the staff who are not adept searchers can play a very important role in these areas.
It always intrigued me that in anything beyond the very simplest search operation, folks with very modest physical skills cold come in quite handy. One example that comes readily to mind - situating someone as an observer in a good spot, giving them some good binos or a spotting scope and a radio.
In the operations in my experience, the presence and participation of the NPS has aided the operation considerably, and they, of course, are not unique in this. I have seen a lot of ranchers provide similar and equally welcome aid. It is just simple humanity.